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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 27
# Folio 9v, page id 00068072:0018
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lu = dx:, 1-
tuitima M
ex eo, e ut sumamus eo pro basi cujus differentialis sit constans, et hinic deducamus
Z et V, quo denique horum op secundum rethodum habeamus X: a, et y: a, ac de
ita problema dende  formabimus tangentrale ex ipsorum X et y notura.
sit ergo q = _ cod: aa et v = aa : co. Eut dz = 3 coco deo: aa, et (t) da
et (gto) dez = 6co dedco: aa et dv (5t0) d = = aa dco: coco et (6to) ddv = aa d codio: c03
 sit jam   (7mo) y: a = sdd Z: dZ et fiet y: a = (6co deo du: 3cocodeo = 2(du: co  y: a
et X: a = (B) od Sadz: do et fiet X: a = - 6co deo (d co : aa): 2a a ld co: co∞) = B s aco. a  X: a
 diffeentia
et quadrando X: a = B - 3 (d : 443) Ex his duobus valoribus formemus aequaationem aliquam
lem rangentialem inter d dy: dx, xa et ya dyid  dy: a 2do: co
+X : a = (13) = 3 c0 d co: a t unde dy : dx = - 2(de: c0): 3  3C03deo: at = 20: 3C0-- (14) dy: dx
Ex 2  90 -2sdo:  et ydeo + cody = edaltcoduo: C1) + 2a dco(fdco: cd) y: 2a
Ex dest yy = 2f 2a y sdco: co fit 2ydy = 2a ydco: c0) +2adu Sdeo: Co seu saco: co = 2ydy -2a 7 acoreo); ratus
Ex gest yeo
cosacvico
Rurjus y ex 8 est yc0 = 2a c0 sdco: co et y deo + codus = 2 a cd0 + 2a deo sdco: 0, et fie t
acqsdcoico
(dco:  = ydco + a dy - 20deo,: 2 adeo, et duos valores aequando  ae quando fet
2 dyde 2 a fdco: de0) fiet (2y dex dio - 2a y dio sitio : c0) = (ydus de) + co dydys - 2a deo de
et pro deo:ce unde yday (deo) - 2 a yu) dco: ) = 2a dudiv) 2 a divlled)
racuiu
quam ut comprobemus explicentur omnia erib y = 200 ady -2ba dco 3: codus -Cdeo = ra absuadum
20 unde evanescit y nec licet y vere habere ex stit istis, aut ex xydu du, ex
nam  transcendes, haberetur ex ordinariis. Ergo mutanda hypothesis ut omnia haberi
ton dot ep l i co
Z=  a(deo: co et d2= ado: c et  =y: a
du = 2a X g ys ) et ddze L3) - 2 addeo: coco, et adz: d = - 2 a dcococo, :, adio: c0 = - 2 deo: ee  et s daz: d -- 2. 5 duv: cod
queant
= 2adx:9 22-)
Z= a (de: sit  ded constans
dz = da Sdco do = Aco t. a) fdco: c, do ac: coco quaerenda jam do talis, ut (ddz: dx
seu X. a contineat etiam sdco. sit ergo XCo sdco: co et x debet esse = (7)as daZ: dd, ergo
differentiando per 6 et7 fit du +deo sdeo  addz: de seu de = ilco + deo saco: co ubi pro
daque ponendo valorem ex aeq. 3 fiet dv = - 2adcot: c0n):, deo + deo sdeo: c seu dod- -2a A coW0 (1 + (deo . c0)
d costeo- deo +du s)  Pil
Erpo fcos
Co + sdeo sico - co sio
ddq. do = deo + ico ste
et detzue = -2a ded dloico
fiet du-
devaco: coco
et Z: dx = dco, ((1 + sdco: co). a ut in og. 8. Hinc X 2 per 8 el 5 fitp. 9 co. 2
Ex d fit anc e  dco X  d  deo = dx: 1 -(9. 24). Rursus ex 20 est
dd= Bra, _ Cx + XAy,: gy et fiet (2ayydx = qaay dx + q aaxdy  2a yy x - 2axydy
seu raxdy + Kydy + 2ay dx seu dy: d = 2) 2AX +X)  2ay, quod comprobemus
: (1): co): ut sdevico= d  dco: a  Drc (ad sdedici,: 4 ap (dco: )
 seu dy: dx = (19) 2ay: 2aX - xy
ad (do:u :, (2saco ((duo: 9 +  a co (qud) a: 20
to Eacla
AaLC+ C(dco) 4a (1 + sdco: co) 
HH. 5dco: c
a: 1+ Sdco: cd = = a:
lx daque: ido
alculus ergo est ptrobis
antrabetur daz ex. ag. 3.
quaeatur
do ddv per 4 Ehdv- radcor
et ddo= 2a dco  fdo. 10) A (∞7o ac. . id e q(+ Salvico)
prodit utrobique ded inveniamus per 
2dio: cd
et ddx = aa  ua du Deodco (do: C0) (F coco : C0): Cot (2) (1 + sdco : c0)
jam superest ut
jam habetur daque
jan coco (1+ Sdco. co) = - raadeo:  du
et fiet ddv = 2aa codcodeo,: 4acodeo d deodeo: dode seu d da= dvdo cosdadiut, ita
se dav) 2 aa eo deodeo sda: co,: co4 ( (1+ (dco : c0)
et deo = adeo ddz: d z dz fit di 47 = - a a7
dvdv = 4a c: cot A) (1+ sdco : co) d ddque - 2 a deo deo: coco
d dqe cod co= adco: de
racdc).  at: du ) -1
coprobatii sdeo. C aaAtcodu: coc): aaflco dvicoc) rcodo
Ex 9 Sdco: co = szaa dco:  cocd dv) _ - 1. seu (per 2.
seu aCoico a 2 2dq. An  dZ dZ: dzdz dcod
c0 = 22dco aez
(D 1+ fdco: 1+ fdco -2 cocst ad
 ado drocodoicodo et per tollendo e0 per 2. fit
duode (2aadcodeo:  (+ sde))
et cofdeo: c0 adeo :, d2d duodv
=2d3 d7 - dcodx: deodo
: c0 = 2dZdZ + dcodv e,: dcodv. comprobando 2 aa(deo duo :coeo) P-2aadco de (. Coco C + sdlo),
sdcoico = - 2aadco (. dx aasil eodco: d zd2)) - 1 = a ad - 27dZ: dvato
a deo des ilaca dco :l Eoc5) -
Ex y Sdco: co =
= raa deodeo (: coco C 1 + sded : c0)) =  - (1 (:, +
3:+ f. 1 (., 1 + sau = (D saro (+D ut desidem
-2d3dZ- deodvo: deo dx  Ergo ex  n
perro e 2 e  tollendo a ope2 et 17 sollendo a fit 
Ergo Sdeo: co Z
notandum erorem commissum, qui
in pergaa deo: coco. non -2 a dcoda: coco 2Faq
sed aulequa
reassumere cheda plagula peculi nove
pro de in vas aazu
lax= det Henc cosdco: co-
dque: cvi co) seu coc dZe dco: de
in pleraque influst, nam
praestat calculum
Novembr. 1703.
=-2Co de
= - 2 dco : 00
du +alos
seu dv = = raa duo
cocodeo noco
qudx= raxdys. 2aydx
yxdus tygax
A)dZ = aa dio du: coc
conadvin